Did Frodo Fail?
This might seem like a strange place to address this decades-old question, but the answer to it depends very much on whether we look at Frodo from a weak-group or strong-group viewpoint. In fact, much of his post-Quest suffering comes from his own tendency to do things on his own, in particular with regard to the Quest; we might attribute this at least partly to his years of being something of an outsider in a strong group society. His original plan to go into lifelong exile by himself is changed by Gandalf and Sam. After the meeting with Gildor, Frodo again considers leaving everyone - even Sam - behind, but he allows Sam to make the choice. Sam, of course, has already told the Elves that he won't leave Frodo even if 'he climbs to the moon' (which would probably have been an easier journey than the one he ends up making).
The existence of the Conspiracy tells us that Frodo has friends who understand him well enough to know that he is likely to go off on his own and who love him enough to not let him do it. (Merry says they've been concerned about the possibility ever since Bilbo left, so it's been a long vigil - so long, in fact, that Pippin most likely joined it in progress.) Frodo's ultimately relieved response tells us that it's concern for his friends' safety that led him to decide to leave them behind, rather than an actual desire to be isolated. But no matter how much he appreciates his friends' companionship, he's not going to ask for it and, with one notable exception, he never does. Strider invites himself along at Bree. The Fellowship is assembled by Elrond and Gandalf. After the confrontation with Boromir, Frodo once again determines to go ahead on his own; this time it's only Sam who figures things out quickly enough to go with him.
Given this history, it's remarkable that one of the last things we see Frodo do is ask Sam to come with him to the Havens. It's still so difficult for Frodo to make the request that he doesn't tell Sam where they're going. During the final scenes of LotR, we're not in Frodo's point of view; If we were, I suspect we'd find the same kind of internal debate going on as we were privy to in the scene revealing the Conspiracy, with Frodo trying to decide how to tell Sam he's leaving. The meeting with Elrond, Galadriel and Bilbo occurs before he's able to do so.
What does this brief summary say about Frodo's failure or success? I believe it says that we can expect Frodo to see himself as more of a failure than others in the story do. The main question in the "Did Frodo fail?" debate is what, exactly, his task was. Was it to cast the Ring into the fires of Mount Doom? Or was it to get the Ring as far on that path as he was able to, using all the strength and wisdom and wit he had, so that other powers could come into play? Tolkien said it was the latter, but Frodo seems to have thought otherwise, at least subconsciously. At a moment when he's exhausted to the point of collapse, and so more likely to let his deepest-felt truth come out, he says to Faramir, 'I must find the Mountain of Fire and cast the thing into the gulf of Doom. Gandalf said so.' Gandalf had not said so, but it seems that's what Frodo heard. So, going by Frodo's own definition of his task, he failed.
But that's looking at it from a weak-group viewpoint, which isn't where the book as a whole is coming from. LotR is a book written from a strong-group way of looking at things. Not only is there no one, central hero, but nothing vital is ever accomplished by one heroic person acting alone. From armies and alliances between kingdoms, to a woman and a hobbit together fulfilling the prophecy that "no man" will destroy the Witch-king, Middle-earth is saved by joint effort. Each person's role is vitally important, but is bound up with everyone else's vitally important role. If all of the Fellowship were warriors, who would carry the Ring? Yet the Ring-bearer cannot say to the warriors, "I do not need you," any more than the eye can say it to the ear.
If we take Paul's image further, we can see that the only way this could work is if all the participants acted as one, united body, even - or especially - when physically apart. In order for a body to act with any kind of coordination, it needs a head. In Paul's image, Christ is that head. The Third Age is pre-Christian, so the characters have no concept of being the Body of Christ. But for those who give themselves with love for the good of all, "the One never named, whose presence is felt on every page" is guiding each part of the body. (Putting all this coordination into the writing of the interwoven plot was no small feat.) I believe this concept of a pre-Christian Body of Christ is so "absorbed into the story" of The Lord of the Rings that it infuses the entire book and is one of those things that makes it fundamentally a Catholic and religious work rather than a work with some Catholic and religious elements.
Frodo's fate, however, reminds us that not everyone who's part of such a body automatically understands and accepts what it means. Frodo had to take more than one journey to understand what it meant in his life, and so do most of us. We'll spend one section of this book looking at Frodo's spiritual progress, an important part of which was his need to acknowledge - to himself - that he was one part of a body and not a single individual standing on his own.
Looking back on our previous discussion of salvation through faith and works makes me wonder whether there's something in our human nature that makes us want to be able to save ourselves? Is that what Frodo finally had to let go of? When he accepts the opportunity to sail West, he's surrendering to those who might be called the epitome of a strong group: the Valar themselves and, ultimately, the One who is also Three.